Question: Let & = {e1, ez, ey } be the standard basis of IR and let C be the standard basis of My (IR), that is,

Let & = {e1, ez, ey } be the standard basis of IR" and let C be the standard basis of My (IR), that is, Let A E M2 (IR) and consider the linear transformation T : M2(R) R3 Tr(B) B Tr(AB) Tr(A2 B) Suppose 0 0 2 3 14 21 42 (a) (2 points) Compute T ( , using [Tle c. Show you work. (Do not use part (d). (b)(8 points) Find a basis for Ker(7) and a basis for Im(7). Justify your answer. (c) (2 points) What are rank(T) and nullity (T)? Justify your answer. (d) (4 points) Find the matrix A. Justify your answer. (Hint: Let A = and compute Tr(AEij) for each Eij E C.)

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